INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS

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1 INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS NATHAN BROWN, RACHEL FINCK, MATTHEW SPENCER, KRISTOPHER TAPP, AND ZHONGTAO WU Abstract. We classify the left-invariant metrics with nonnegative sectional curvature on SO(3) and U(2). 1. Introduction Every compact Lie group admits a bi-invariant metric, which has nonnegative sectional curvature. This observation is the starting point of almost all known constructions of nonnegatively and positively curved manifolds. The only exceptions are found in [1], [6], and [12], and many of these exceptions admit a different metric which does come from a bi-invariant metric on a Lie group; see [10]. In order to generalize this cornerstone starting point, we consider the following problem: for each compact Lie group G, classify the left-invariant metrics on G which have nonnegative sectional curvature. The problem is well-motivated because each such metric, g l, provides a new starting point for several known constructions. For example, suppose H G is a closed subgroup. The left-action of H on (G, g l ) is by isometries. The quotient space H\G inherits a metric of nonnegative curvature, which is generally inhomogeneous. Geroch proved in [5] that this quotient metric can only have positive curvature if H\G admits a normal homogeneous metric with positive curvature, so new examples of positive curvature cannot be found among such metrics. However, it seems worthwhile to search for new examples with quasi- or almost-positive curvature (meaning positive curvature on an open set, or on an open and dense set). A second type of example occurs when H G is a closed subgroup, and F is a compact Riemannian manifold of nonnegative curvature on which H acts isometrically. Then H acts diagonally on (G, g l ) F. The quotient, M = H\(G F ), inherits a metric with nonnegative curvature. M is the total space of a homogeneous F -bundle over H\G. Examples of this type with quasi-positive curvature are given in [9]. It is not known whether new positively curved examples of this type could exist. Date: August 25, Mathematics Subject Classification. 53C. Supported in part by NSF grant DMS and NSF grant DMS

2 2 NATHAN BROWN, RACHEL FINCK, MATTHEW SPENCER, KRISTOPHER TAPP, AND ZHONGTAO WU In this paper, we explore the problem for small-dimensional Lie groups. In section 3, we classify the left-invariant metrics on SO(3) with nonnegative curvature. This case is very special because the bi-invariant metric has positive curvature, and therefore so does any nearby metric. The solution is straightforward and known to some, but does not appear in the literature. In section 4, we classify the left-invariant metrics on U(2) with nonnegative curvature. This classification does not quite reduce to the previous, since not all such metrics lift to product metrics on U(2) s double-cover S 1 SU(2). The only examples in the literature of left-invariant metrics with nonnegative curvature on compact Lie groups are obtained from a bi-invariant metric by shrinking along a subalgebra (or a sequence of nested subalgebras). We review this construction in section 5. Do all examples arise in this way? In section 6, we outline a few ways to make this question precise. Even for U(2), the answer is essentially no. Also, since each of our metrics on U(2) induces via Cheeger s method a left-invariant metric with nonnegative curvature on any compact Lie group which contains a subgroup isomorphic to U(2), our classification for small-dimensional groups informs the question for large-dimensional groups. The fourth author is pleased to thank Wolfgang Ziller for suggesting Proposition 3.1 and for valuable feedback on the paper. 2. Background Let G be a compact Lie group with Lie algebra g. Let g 0 be a bi-invariant metric on G. Let g l be a left-invariant metric on G. The values of g 0 and g l at the identity are inner products on g which we denote as, 0 and, l. We can always express the latter in terms of the former: A, B l = φ(a), B 0 for some positive-definite self-adjoint φ : g g. The eigenvalues and eigenvectors of φ are called eigenvalues and eigenvectors vectors of the metric g l. There are several existing formulas for the curvature tensor of g l at the identity, which we denote as R : g g g g. Püttmann s formula from [8] says that for all x, y, z, w g, (2.1) R(x, y)z, w l = 1 4 ( [φx, y], [z, w] 0 + [x, φy], [z, w] 0 + [x, y], [φz, w] 0 + [x, y], [z, φw] 0 ) 1 4 ( [x, w], [y, z] l [x, z], [y, w] l 2 [x, y], [z, w] l ) ( B(x, w), φ 1 B(y, z) 0 B(x, z), φ 1 B(y, w) 0 ), where B is defined by B(x, y) = 1 2 ([x, φy] + [y, φx]). Sectional curvatures can be calculated from this formula. An alternative formula by Milnor for the sectional curvatures of g l is found in [7]. If {e 1,..., e n } g is a g l -orthonormal basis,

6 6 NATHAN BROWN, RACHEL FINCK, MATTHEW SPENCER, KRISTOPHER TAPP, AND ZHONGTAO WU Here we assume for convenience that g 0 is scaled such that the Lie bracket structure of the SU(2) factor is given by equation 3.2. The lemma follows by inspection, since ad E0 is skewadjoint if and only if α 0ij = α 0ji for all i, j. The next lemma applies more generally to any left-invariant metric on any Lie group. Lemma 4.3. The left-invariant vector field x g is parallel if and only if ad x is skew-adjoint and x, [g, g] l = 0. Proof. For all y, z g, (4.1) 2 y, z x l = z, [y, x] l + x, [y, z] l y, [x, z] l. If ad x is skew-adjoint, then first and last terms of 4.1 sum to 0. If additionally x, [g, g] l = 0, then the middle term is also 0. So z x = 0 for all z g, which means that x is parallel. Conversely, assume that x is parallel, so the left side of 4.1 equals 0 for all y, z g. When y = z, this yields 2 y, [y, x] l = 0 for all y g, which implies that ad x is skew-adjoint. This property makes the first and third terms of 4.1 sum to zero, so x, [y, z] l = 0 for all y, z g. In other words, x, [g, g] l = 0. The next lemma from [7] also applies more generally to any left-invariant metric on any Lie group. We use r to denote the Ricci curvature of g l. Lemma 4.4 (Milnor). If x g is g l -orthogonal to the commutator ideal [g, g], then r(x) 0, with equality if and only if ad x is skew-adjoint with respect to g l. Proof of Proposition 4.1. Suppose that g l has nonnegative curvature. Then ad E0 is skewadjoint by Milnor s Lemma 4.4. Next, Lemma 4.2 implies that one of the three conditions of the proposition hold. It remains to prove that {λ 1, λ 2, λ 3 } satisfy the constraints for eigenvalues of a nonnegatively curved metric on SO(3) (or equivalently on SU(2)). By Lemma 4.3, E 0 is parallel. Using Equation 4.1, this implies that the SU(2) factor of G is totally geodesic, so its induced metric has nonnegative curvature, which gives the constraints on the λ s. Conversely, suppose {λ 1, λ 2, λ 3 } satisfies the constraints for eigenvalues of a nonnegatively curved metric on SU(2), and that one of the three conditions of the proposition holds. By Lemma 4.2, ad E0 is skew-adjoint, so by Lemma 4.3, E 0 is parallel. This implies that the SU(2) factor of G is totally geodesic. It has nonnegative curvature because of the constraints on the λ s. Consider the curvature operator of g l expressed in the following basis of 2 (g): {E 0 e 1, E 0 e 2, E 0 e 3, e 1 e 2, e 2 e 3, e 3 e 1 }. Since E 0 is parallel, R(E 0 e i ) = 0. Furthermore, R(e i e j ) is calculated in the totally geodesic SU(2). It follows that the curvature operator is nonnegative. We refer to metrics of types (2) and (3) in proposition 4.1 as twisted metrics. We know from the splitting theorem that nonnegatively curved twisted metrics are locally isometric to

8 8 NATHAN BROWN, RACHEL FINCK, MATTHEW SPENCER, KRISTOPHER TAPP, AND ZHONGTAO WU 5. Review of Cheeger s method In the literature, the only examples of left-invariant metrics with nonnegative curvature on compact Lie groups come from a construction which, in its greatest generality, is due to Cheeger [1]. In this section, we review Cheeger s method. Let H G be compact Lie groups with Lie algebras h g. Let g 0 be a left-invariant and Ad H -invariant metric on G with nonnegative curvature. Let g H be a right-invariant metric on H with nonnegative curvature. The right diagonal action of H on (G, g 0 ) (H, g H ) is by isometries. The quotient is diffeomorphic to G and inherits a submersion metric, g 1, which is left-invariant and has nonnegative curvature: (G, g 1 ) = ((G, g 0 ) (H, g H ))/H. If g H is bi-invariant, then g 1 is Ad H -invariant. The metrics g 0 and g 1 agree orthogonal to h. To describe g 1 on h, let {e 1,..., e k } denote a g H -orthonormal basis of h. Let A and Ã respectively denote the k-by-k matrices whose entries are a ij = g 0 (e i, e j ) and ã ij = g 1 (e i, e j ). The restrictions of g 1 and g 0 to h are shown in [1] to be related by the equation: (5.1) Ã = A(I + A) 1. A common special case occurs when g 0 is bi-invariant, and g H is a multiple, λ, of the restriction of g 0 to H. In this case, which was first studied in [4], g 1 restricted to h is obtained from g 0 by scaling by the factor t = λ/(1+λ). In this case, g 1 is Ad H -invariant, and is therefore Ad K -invariant for any closed subgroup K H. So Cheeger s method can be applied again: (G, g 2 ) = ((G, g 1 ) (K, g K ))/K, and so on. Iterating Cheeger s method in this was is studied in [3]. In summary, whenever H 0 H 1 H l 1 H l = G is a chain of closed subgroups of G with Lie algebras h 0 h 1 h l 1 h l = g, one can apply Cheeger s method l-times. One chooses a starting bi-invariant metric g 0 on G and l constants {λ 0,..., λ l 1 }. The result is a new left-invariant metric with nonnegative curvature on G. The sub-algebras h i are separately scaled by factors determined by the λ s, with h i scaled a greater amount than h i+1. More precisely, the eigenvalues of the metric are a strictly increasing sequence t 0 < t 1 < < t l 1 < t l = 1. The λ s can be chosen to give any such strictly increasing sequence. The eigenspace of t 0 equals h 0. For i > 1, the eigenspace of t i equals the g 0 -orthogonal compliment of h i 1 in h i. 6. Do all metrics come from Cheeger s method? Let G be a compact Lie group. Does every left-invariant metric on G with nonnegative curvature come from Cheeger s method? In this section, we outline several ways to precisely formulate this question, and address the cases G = SO(3) and G = U(2). First, one might ask whether all examples arise by starting with a bi-invariant metric g 0 on G and applying Cheeger s method to a chain of subgroups H 0 H l 1 G, each time

9 INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS 9 choosing the metric on H i to be a multiple, λ i, of the restriction of g 0 to H i. The answer is clearly no, since not all metrics on SO(3) arise in this fashion. But SO(3) is a very special case, since its bi-invariant metric has positive curvature. There are several ways to modify the question to reflect the guess that SO(3) is the only exception. What if, in each application of Cheeger s method, one allows more general right-invariant metrics on the H i s? For example, if H i = SO(3) or SU(2), we allow any right-invariant metric with nonnegative curvature. With this added generality, the answer is still no, since metrics on SO(3) which are nonnegatively but not positively curved do not arise in this fashion: Proposition 6.1. If g 0 is a bi-invariant metric on SO(3) and g R is a right-invariant metric with nonnegative curvature on SO(3), then the following has strictly positive curvature: (SO(3), g l ) := ((SO(3), g 0 ) (SO(3), g R ))/SO(3). Further, every positively-curved left-invariant metric g l can be described in this way for some bi-invariant metric g 0 and some positively curved right-invariant metric g R. Proof. Let π : (SO(3), g 0 ) (SO(3), g R ) (SO(3), g l ) denote the projection, which is a Riemannian submersion. Let {e 1, e 2, e 3 } so(3) be a g 0 -orthonormal basis of eigenvectors of the metric g R, with eigenvalues denoted {λ 1, λ 2, λ 3 }. The horizontal space of π at the identity is the following subspace of so(3) so(3): (6.1) H (e,e) = {(V, W ) (V, W ), (e i, e i ) = 0 for all i = 1, 2, 3} = {(V, W ) V, e i 0 + λ i W, e i 0 = 0 for all i = 1, 2, 3}. Now let X 1 = (V 1, W 1 ), X 2 = (V 2, W 2 ) H (e,e) be linearly independent vectors. From equation 6.1, we see that V 1, V 2 are linearly independent as well. Since (SO(3), g 0 ) has positive curvature, κ(v 1, V 2 ) > 0, so κ(x 1, X 2 ) > 0. Finally, O Neill s formula implies that (SO(3), g l ) has strictly positive curvature. To prove the second statement of the proposition, notice that by equation 5.1, the eigenvalues { λ 1, λ 2, λ 3 } of the metric g l are determined from the eigenvalues {λ 1, λ 2, λ 3 } of the metric g R as follows: (6.2) λi = λ i 1 + λ i for each i = 1, 2, 3. Suppose that ( λ 1, λ 2 ) is an arbitrary pair such that the triplet { λ 1, λ 2, 1} strictly satisfies the inequalities of Proposition 3.1. Define: λ 1 := a λ a(1 λ 1 ), λ 2 := a λ a(1 λ 2 ), λ 3 := a. For small enough a, the triplet {λ 1, λ 2, λ 3 } strictly satisfies the inequalities of Proposition 3.1. This is because, when scaled such that the third equals 1, the triplet approaches { λ 1, λ 2, 1} as

10 10 NATHAN BROWN, RACHEL FINCK, MATTHEW SPENCER, KRISTOPHER TAPP, AND ZHONGTAO WU a 0. If g R has eigenvalues {λ 1, λ 2, λ 3 }, then by equation 6.2, g l has the following eigenvalues: {( ) ( ) ( )} a a a λ 1, λ 2,. 1 + a 1 + a 1 + a Further, g R has positive curvature because the inner products which extend to right-invariant metrics with positive curvature are the same ones that extend to left-invariant metrics of positive curvature. This is because a a 1 is an isometry between left and right-invariant metrics determined by the same inner product. Thus, g R can be chosen such that g l is a multiple of any prescribed positively curved metric. Then, by scaling the bi-invariant metric g 0, this multiple can be made to be 1. Starting with a bi-invariant metric g 0 on G = SU(2) U(1) and applying Cheeger s method, one cannot obtain product metrics for which the SU(2)-factor has nonnegative but not positive curvature, nor can one obtain twisted metrics of type (2). Notice that the only chains of increasing-dimension subgroups are U(1) G (possibly embedded diagonally), T 2 G (any maximal torus), U(1) T 2 G, and SU(2) G. To obtain more metrics, in addition to allowing general right-invariant metrics on the H i s as above, one could allow a more general starting metric g 0. For example, when G = SU(2) U(1), one could allow g 0 to be any left-invariant product metric with nonnegative curvature. Even with this added generality, it is straightforward to see that one cannot obtain twisted metrics of type (3) for which the totally geodesic SU(2) has nonnegative but not positive curvature. References 1. J. Cheeger, Some examples of manifolds of nonnegative curvature, J. Differential Geom. 8 (1972), J. Cheeger and D. Gromoll, On the structure of complete open manifolds of nonnegative curvature, Ann. of Math. 96 (1972), W. Dickinson, Curvature properties of the positively curved Eschenburg spaces, Differential Geom. Appl. 20 (2004), no. 1, J.H. Eschenburg, Inhomogeneous spaces of positive curvature, Differential Geom. Appl. 2 (1992), R. Geroch, Group-quotients with positive sectional curvatures, Proc. of the Amer. Math. Soc. 66 (1977), no. 2, K. Grove and W. Ziller, Curvature and symmetry of Milnor spheres, Ann. of Math. 152 (2000), J. Milnor, Curvatures of left-invariant metrics on lie groups, Advances in Math. 21 (1976), T. Püttmann, Optimal pinching constants of odd dimensional homogeneous spaces, Ph. D. thesis, Ruhr- Universität, Germany, K. Tapp, Quasi-positive curvature on homogeneous bundles, J. Differential Geom. 66 (2003), B. Totaro, Cheeger manifolds and the classification of biquotients, J. Differential Geom. 61 (2002), no. 3, N. Wallach, Compact Riemannian manifolds with strictly positive curvature, Ann. of Math. 96 (1972), D. Yang, On complete metrics of nonnegative curvature on 2-plane bundles, Pacific J. of Math. 171, No. 2, 1995.

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